Question

For a closed cylinder with radius r ⁢ cm and height h ⁢ cm, find the dimensions giving the minimum surface area, given that the volume is 36 cm3.

Answer 1

Answer:

The radius is 1.79 cm and height is 3.58 cm.

Step-by-step explanation:

The surface area of the closed cylinder is given by,

$S = 2\pi R^{2} + 2\pi Rh$

And volume is given by,

$V = \pi R^{2}h$

Where, R is radius and h is height of cylinder.

Volume is given to be  36 cm3.

$V = \pi R^{2}h = 36$

$h = \frac{36}{\pi R^{2}}$

Inserting this value of h in Surface area equation, we get,

$S = 2\pi R^{2} + 2\pi R(\frac{36}{\pi R^{2}})$

$S = 2\pi R^{2} + \frac{72}{R}$

Now differentiating wrt x to find minimum, inserting $\frac{dS}{dR} = 0$, we get,

$4\pi R = \frac{72}{R^{2}}$

$R = (\frac{18}{\pi })^{\frac{1}{3}} = 1.79 cm$

Thus, h = 3.58 cm from above equation.